This lively collection of essays examines in witty detail the history of some of the concepts involved in bringing statistical argument "to the table," and some of the pitfalls that have been encountered. The topics range from seventeenth-century medicine and the circulation of blood, to the cause of the Great Depression and the effect of the California gold discoveries of 1848 upon price levels, to the determinations of the shape of the Earth and the speed of light, to the meter of Virgil's poetry and the prediction of the Second Coming of Christ. The title essay tells how the statistician Karl Pearson came to issue the challenge to put "statistics on the table" to the economists Marshall, Keynes, and Pigou in 1911. The 1911 dispute involved the effect of parental alcoholism upon children, but the challenge is general and timeless: important arguments require evidence, and quantitative evidence requires statistical evaluation. Some essays examine deep and subtle statistical ideas such as the aggregation and regression paradoxes; others tell of the origin of the Average Man and the evaluation of fingerprints as a forerunner of the use of DNA in forensic science. Several of the essays are entirely nontechnical; all examine statistical ideas with an ironic eye for their essence and what their history can tell us about current disputes.

In a series of 50 accessible essays, Tony Crilly explains and introduces the mathematical laws and principles - ancient and modern, theoretical and practical, everyday and esoteric - that allow us to understand the world around us. From Pascal's triangle to money management, ideas of relativity to the very real uses of imaginary numbers, 50 Maths Ideas is a complete introduction to the most important mathematical concepts in history.

This book provides a complete understanding of chaotic dynamics in mathematics, physics, and the real world, with an explanation of why it is important and how it differs from the idea of randomness. The author draws on certain physical systems and phenomena, for example the weather forecast, a pendulumn, a coin toss, mass transit, politics, and the role of chaos in in gambling and the stock-market.

This book describes in detail the various theories on the shape of the Earth from classical antiquity to the present day and examines how measurements of its form and dimensions have evolved throughout this period. The origins of the notion of the sphericity of the Earth are explained, dating back to Eratosthenes and beyond, and detailed attention is paid to the struggle to establish key discoveries as part of the cultural heritage of humanity. In this context, the roles played by the Catholic Church and the philosophers of the Middle Ages are scrutinized. Later contributions by such luminaries as Richer, Newton, Clairaut, Maupertuis, and Delambre are thoroughly reviewed, with exploration of the importance of mathematics in their geodetic enterprises. The culmination of progress in scientific research is the recognition that the reference figure is not a sphere but rather a geoid and that the earth's shape is oblate. Today, satellite geodesy permits the solution of geodetic problems by means of precise measurements. Narrating this fascinating story from the very beginning not only casts light on our emerging understanding of the figure of the Earth but also offers profound insights into the broader evolution of human thought.

This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis--finding the "right axioms" to prove fundamental theorems--and giving a novel approach to logic. Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the "right axiom" to prove it. By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics.

This is the first in a series of volumes, which provide an extensive overview of conjectures and open problems in graph theory. The readership of each volume is geared toward graduate students who may be searching for research ideas. However, the well-established mathematician will find the overall exposition engaging and enlightening. Each chapter, presented in a story-telling style, includes more than a simple collection of results on a particular topic. Each contribution conveys the history, evolution, and techniques used to solve the authors' favorite conjectures and open problems, enhancing the reader's overall comprehension and enthusiasm. The editors were inspired to create these volumes by the popular and well attended special sessions, entitled "My Favorite Graph Theory Conjectures," which were held at the winter AMS/MAA Joint Meeting in Boston (January, 2012), the SIAM Conference on Discrete Mathematics in Halifax (June,2012) and the winter AMS/MAA Joint meeting in Baltimore(January, 2014). In an effort to aid in the creation and dissemination of open problems, which is crucial to the growth and development of a field, the editors requested the speakers, as well as notable experts in graph theory, to contribute to these volumes.

This book presents a detailed description of the development of statistical theory. In the mid twentieth century, the development of mathematical statistics underwent an enduring change, due to the advent of more refined mathematical tools. New concepts like sufficiency, superefficiency, adaptivity etc. motivated scholars to reflect upon the interpretation of mathematical concepts in terms of their real-world relevance. Questions concerning the optimality of estimators, for instance, had remained unanswered for decades, because a meaningful concept of optimality (based on the regularity of the estimators, the representation of their limit distribution and assertions about their concentration by means of Anderson's Theorem) was not yet available. The rapidly developing asymptotic theory provided approximate answers to questions for which non-asymptotic theory had found no satisfying solutions. In four engaging essays, this book presents a detailed description of how the use of mathematical methods stimulated the development of a statistical theory. Primarily focused on methodology, questionable proofs and neglected questions of priority, the book offers an intriguing resource for researchers in theoretical statistics, and can also serve as a textbook for advanced courses in statisticc.

Major shifts in the field of model theory in the twentieth century have seen the development of new tools, methods, and motivations for mathematicians and philosophers. In this book, John T. Baldwin places the revolution in its historical context from the ancient Greeks to the last century, argues for local rather than global foundations for mathematics, and provides philosophical viewpoints on the importance of modern model theory for both understanding and undertaking mathematical practice. The volume also addresses the impact of model theory on contemporary algebraic geometry, number theory, combinatorics, and differential equations. This comprehensive and detailed book will interest logicians and mathematicians as well as those working on the history and philosophy of mathematics.

This book presents William Clifford's English translation of Bernhard Riemann's classic text together with detailed mathematical, historical and philosophical commentary. The basic concepts and ideas, as well as their mathematical background, are provided, putting Riemann's reasoning into the more general and systematic perspective achieved by later mathematicians and physicists (including Helmholtz, Ricci, Weyl, and Einstein) on the basis of his seminal ideas. Following a historical introduction that positions Riemann's work in the context of his times, the history of the concept of space in philosophy, physics and mathematics is systematically presented. A subsequent chapter on the reception and influence of the text accompanies the reader from Riemann's times to contemporary research. Not only mathematicians and historians of the mathematical sciences, but also readers from other disciplines or those with an interest in physics or philosophy will find this work both appealing and insightful.

Discrete mathematics has been rising in prominence in the past fifty years, both as a tool with practical applications and as a source of new and interesting mathematics. The topics in discrete mathematics have become so well developed that it is easy to forget that common threads connect the different areas, and it is through discovering and using these connections that progress is often made. For over fifty years, Ron Graham has been able to illuminate some of these connections and has helped to bring the field of discrete mathematics to where it is today. To celebrate his contribution, this volume brings together many of the best researchers working in discrete mathematics, including Fan Chung, Erik D. Demaine, Persi Diaconis, Peter Frankl, Alfred W. Hales, Jeffrey C. Lagarias, Allen Knutson, Janos Pach, Carl Pomerance, N. J. A. Sloane, and of course, Ron Graham himself.

In the mid-eighteenth century, Swiss-born mathematician Leonhard Euler developed a formula so innovative and complex that it continues to inspire research, discussion, and even the occasional limerick. Dr. Euler's Fabulous Formula shares the fascinating story of this groundbreaking formula--long regarded as the gold standard for mathematical beauty--and shows why it still lies at the heart of complex number theory. In some ways a sequel to Nahin's An Imaginary Tale, this book examines the many applications of complex numbers alongside intriguing stories from the history of mathematics. Dr. Euler's Fabulous Formula is accessible to any reader familiar with calculus and differential equations, and promises to inspire mathematicians for years to come.

Newton's Principia is perhaps the second most famous work of mathematics, after Euclid's Elements. Originally published in 1687, it gave the first systematic account of the fundamental concepts of dynamics, as well as three beautiful derivations of Newton's law of gravitation from Kepler's laws of planetary motion. As a book of great insight and ingenuity, it has raised our understanding of the power of mathematics more than any other work. This heavily annotated translation of the third and final edition (1726) of the Principia will enable any reader with a good understanding of elementary mathematics to easily grasp the meaning of the text, either from the translation itself or from the notes, and to appreciate some of its significance. All forward references are given to illuminate the structure and unity of the whole, and to clarify the parts. The mathematical prerequisites for understanding Newton's arguments are given in a brief appendix.

In this book, Johnny Ball tells one of the most important stories in world history - the story of mathematics.

By introducing us to the major characters and leading us through many historical twists and turns, Johnny slowly unravels the tale of how humanity built up a knowledge and understanding of shapes, numbers and patterns from ancient times, a story that leads directly to the technological wonderland we live in today. As Galileo said, 'Everything in the universe is written in the language of mathematics', and Wonders Beyond Numbers is your guide to this language.

Mathematics is only one part of this rich and varied tale; we meet many fascinating personalities along the way, such as a mathematician who everyone has heard of but who may not have existed; a Greek philosopher who made so many mistakes that many wanted his books destroyed; a mathematical artist who built the largest masonry dome on earth, which builders had previously declared impossible; a world-renowned painter who discovered mathematics and decided he could no longer stand the sight of a brush; and a philosopher who lost his head, but only after he had died.

Enriched with tales of colourful personalities and remarkable discoveries, there is also plenty of mathematics for keen readers to get stuck into. Written in Johnny Ball's characteristically light-hearted and engaging style, this book is packed with historical insight and mathematical marvels; join Johnny and uncover the wonders found beyond the numbers.

This thorough work presents the fundamental results of modular function theory as developed during the nineteenth and early-twentieth centuries. It features beautiful formulas and derives them using skillful and ingenious manipulations, especially classical methods often overlooked today. Starting with the work of Gauss, Abel, and Jacobi, the book then discusses the attempt by Dedekind to construct a theory of modular functions independent of elliptic functions. The latter part of the book explains how Hurwitz completed this task and includes one of Hurwitz's landmark papers, translated by the author, and delves into the work of Ramanujan, Mordell, and Hecke. For graduate students and experts in modular forms, this book demonstrates the relevance of these original sources and thereby provides the reader with new insights into contemporary work in this area.

This book is about the rise and supposed fall of the mean value theorem. It discusses the evolution of the theorem and the concepts behind it, how the theorem relates to other fundamental results in calculus, and modern re-evaluations of its role in the standard calculus course. The mean value theorem is one of the central results of calculus. It was called "the fundamental theorem of the differential calculus" because of its power to provide simple and rigorous proofs of basic results encountered in a first-year course in calculus. In mathematical terms, the book is a thorough treatment of this theorem and some related results in the field; in historical terms, it is not a history of calculus or mathematics, but a case study in both. MVT: A Most Valuable Theorem is aimed at those who teach calculus, especially those setting out to do so for the first time. It is also accessible to anyone who has finished the first semester of the standard course in the subject and will be of interest to undergraduate mathematics majors as well as graduate students. Unlike other books, the present monograph treats the mathematical and historical aspects in equal measure, providing detailed and rigorous proofs of the mathematical results and even including original source material presenting the flavour of the history.

Karl Menger, one of the founders of dimension theory, is among the most original mathematicians and thinkers of the twentieth century. He was a member of the Vienna Circle and the founder of its mathematical equivalent, the Viennese Mathematical Colloquium. Both during his early years in Vienna and, after his emigration, in the United States, Karl Menger made significant contributions to a wide variety of mathematical fields, and greatly influenced many of his colleagues. These two volumes contain Menger's major mathematical papers, based on his own selection from his extensive writings. They deal with topics as diverse as topology, geometry, analysis and algebra, and also include material on economics, sociology, logic and philosophy. The Selecta Mathematica is a monument to the diversity and originality of Menger's ideas.

Originally published in 1921, this book was written by the renowned British mathematician E. H. Neville (1889-1961). The text constitutes an attempt to develop geometrical methods in four-dimensional space. This book will be of value to anyone with an interest in the works of Neville, geometry and the history of mathematics.

The logician Kurt Goedel (1906-1978) published a paper in 1931 formulating what have come to be known as his 'incompleteness theorems', which prove, among other things, that within any formal system with resources sufficient to code arithmetic, questions exist which are neither provable nor disprovable on the basis of the axioms which define the system. These are among the most celebrated results in logic today. In this volume, leading philosophers and mathematicians assess important aspects of Goedel's work on the foundations and philosophy of mathematics. Their essays explore almost every aspect of Godel's intellectual legacy including his concepts of intuition and analyticity, the Completeness Theorem, the set-theoretic multiverse, and the state of mathematical logic today. This groundbreaking volume will be invaluable to students, historians, logicians and philosophers of mathematics who wish to understand the current thinking on these issues.

How can we be sure that Pythagoras's theorem is really true? Why is the 'angle in a semicircle' always 90 degrees? And how can tangents help determine the speed of a bullet? David Acheson takes the reader on a highly illustrated tour through the history of geometry, from ancient Greece to the present day. He emphasizes throughout elegant deduction and practical applications, and argues that geometry can offer the quickest route to the whole spirit of mathematics at its best. Along the way, we encounter the quirky and the unexpected, meet the great personalities involved, and uncover some of the loveliest surprises in mathematics.

This volume of the Mathematics and Culture series is dedicated to Italian artist Armando Pizzicato. The work of Pollock is also discussed, thanks to the collaboration of the Venice Guggenheim Collection. Mathematics creates beauty in architecture, from topology to the projects of Gehry and Piano to the muqarnas of Islam. But mathematics can also save lives. Mathematical models can help fight cancer and AIDS, and intervene in their cure. Mathematics can also be used to protect the environment. In the wake of the devastation caused by Hurricane Katrina, the discussion of the use of mathematical models in meteorology and the prediction of storms is particularly timely. This volume wouldn't be complete without Venice and its glass, the fantastic collection of murrine of Giovanni Sarpellon. There is even a bit of magic, thanks to Bustric. And finally, an homage to a great mathematician: H.S.M. Donald Coxeter.

Gaudi's Sagrada Familia in Barcelona, the America's Cup victory of the Swiss sail boat Alinghi, films such as A Beautiful Mind and Enigma, the theatrical play based on the life of Galois, the images of the Dutch graphic artist M. C. Escher, crypotography, comics. What do all these have in common? Mathematics, of course. Many recent films, plays and books have explored mathematical themes. This book, whose aim is the popularization of mathematics, comprises a series of accounts which demonstrate in these cultural contexts that mathematics is a vibrant and intriguing subject for the non-specialist. In the same spirit, other contributions cover mathematics in art, architecture, cryptography, and the medical and physical sciences.

Originally published in 1962, as the second edition of a 1930 original, 'the main purpose of the book is to give a logical connected account of the subject, by starting with the definition of 'Number' and proceeding in what appears ... to be a natural sequence of steps'. The chapters cover all of the cornerstones of complex mathematical analyses; chapters include, 'Bounds and limits of sequences', 'Integral calculus' and 'Functions of more than one variable'. Multiple examples are included at the end of every chapter to support and illustrate the fundamental concepts; 'I have aimed at presenting the subject in such a way as to make every important concept clearly understood'. Primarily aimed at undergraduates with a background in advanced calculus for study and practice, this comprehensive and dynamic textbook will be of considerable value to scholars of mathematics as well as to anyone with an interest in the history of education.

Originally published in 1926, this book was written to provide mathematical and scientific students with an introduction to the subject of integral calculus. The text was largely planned around the syllabus for the Higher Certificate Examination. A short historical survey is included. This book will be of value to anyone with an interest in integral calculus, mathematics and the history of education.